Students at the figurative
counting stage
143
144
Students at the
figurative counting stage
Students using figurative strategies to aid counting are able to count
collections which are totally or partially concealed. They do not need to
see, feel or hear the items in a collection to be able to count the collection.
However, typically they rely on the simple strategy of counting by ones,
starting from one to find a total. For example, when asked to count a
collection of six items and a collection of three items which are both
concealed from view, the student would be able to count the nine items
often using fingers to represent the items. At the figurative counting stage
students have an understanding of numbers as entities. For example, to
form six using fingers, a student at this stage will raise six fingers
simultaneously. By comparison, at the perceptual counting stage, a student
will construct six using fingers by raising them one at a time as they
count to six. The total of six and three, however, is still found at the
figurative stage by counting from one.
Students at this stage demonstrate an understanding of the conservation
of number and can represent a number in a variety of ways through such
strategies as separating (partitioning) and combining.
Students are able to consistently produce the correct forward and backward
number word sequences in the range from zero to twenty. Many may be
able to go beyond this range. They are able to use their knowledge of
forward and backward number word sequences to produce the number
before and the number after a nominated number.
Students demonstrate their understanding of the numerical value of
numbers by naming numerals and labelling collections of up to twenty
items.
Students at the figurative counting stage are working towards
•
using counting on to solve addition tasks
•
using counting down to solve subtraction tasks
•
developing base ten knowledge
•
forming equal groups and finding their total.
145
FIGURATIVE
The nature of the learner
Developing efficient numeracy strategies
Teaching considerations
When developing teaching and learning programs for students working
at the figurative counting stage, teachers need to consider:
•
Strategy development
Students working at the figurative stage of counting are unable to draw
from a wide range of strategies to solve number problems. Therefore
there is a need to model effective strategies. At this stage the main focus
for teaching should be on developing the counting on procedure. This
will also be a major indicator when assessing students’ abilities.
FIGURATIVE
Ensure that students are given opportunities to practise the strategy of
counting on in small-group, pair and individual activities. Be aware that
students who demonstrate the ability to use counting on may revert to
the strategy of counting from one when they are presented with difficult
tasks.
146
•
Language development
As students develop additional problem-solving strategies, ensure that
they are taught the explicit mathematical language needed to explain
their solutions. Vocalising the procedures used enables students to clarify
their thinking and reinforces concepts.
•
Numeral identification
Whilst using the arithmetical strategies associated with the figurative
stage, students may exhibit different levels of knowledge of numerals.
The identification of numerals does not develop uniformly. Students may
be able to identify some of the numerals beyond twenty before they can
identify all the numerals up to twenty.
Index of activities
Activities which develop base ten knowledge
Collections
Bundling
Trading game
Counting on
Popsticks
Flip and see
149
151
151
151
153
155
Activities which develop counting on and addition
Bucket count on
157
Activities which develop counting on strategies and partitioning
Blocks on a bowl
159
Activities which develop numeral identification and addition
Hundred chart
Add two dice
Posting counters
Two-dice toss
Teddy tummies: addition
Friends of ten
Fishing addition
Using technology
161
163
171
171
173
175
177
179
Activities which develop addition
Rabbits’ ears
Doubles
Doubles plus one
167
167
167
Activities which develop addition and base ten knowledge
Dot pattern cards
169
Activities which develop addition and subtraction
Ring that bell
How many eggs?
Ten frames
181
183
185
Activities which develop subtraction
Subtraction teddies
187
Activities which develop multiplication and division
Arrays
Arrays: changing groups
Guess my square
Triangle teddies
Units for two
Handprints
Turning arrays
Calculating groups
What’s in a square?
Groupies
Dice and grid game
189
189
191
193
195
195
197
199
199
201
203
FIGURATIVE
Page
147
Developing efficient numeracy strategies
Where are they now?
FIGURATIVE
Students do not see ten as a countable
unit. The student’s focus is on the
individual items that form a unit of ten.
In addition or subtraction tasks involving
tens, students count forwards and
backwards by ones.
Where to next?
Students see ten as a unit composed of
ten ones and are able to use the unit to
count. However, they are dependent on
visual representations of ten, such as
strips of card showing ten dots or using
ten fingers.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained
LFN reference
Base ten: level 1.
148
How?
Collections
FIGURATIVE
Present students with a large collection of items, such as counters, pebbles,
or buttons, a supply of containers, such as patty papers or cups, and a
large sheet of cardboard. They will also need two sets of numeral cards
ranging from zero to nine. Divide the chart into a “tens” and a “ones”
column. Present the collection of items to the students and allow them
to count the items. Each time ten items are collected, the students place
the items into a container and move the container to the left-hand side of
the chart, that is, onto the “tens” column. Students then place a numeral
card above the “tens” column, indicating how many groups of ten have
been collected. As succeeding tens are collected, students continue to
add them to the left-hand side of the chart and replace the numeral card
accordingly. Remaining items are placed on the right-hand side of the
chart, in the “ones” column. Students then place a corresponding numeral
card above the “ones” column to form a two-digit number.
Why?
Developing an understanding of tens and ones will assist students in using
strategies other than counting by ones to solve problems.
149
Developing efficient numeracy strategies
Where are they now?
FIGURATIVE
Students do not see ten as a countable
unit. The student’s focus is on the
individual items that form a unit of ten.
In addition or subtraction tasks involving
tens, students count forwards and
backwards by ones.
Where to next?
Students see ten as a unit composed of
ten ones and are able to use the unit to
count. However, they are dependent on
visual representations of ten, such as
strips of card showing ten dots or using
ten fingers.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Base ten: level 1.
150
How?
Bundling
Present the students with large collections of popsticks. Have the students
bundle the popsticks into groups of ten and place any remaining popsticks
to the side of the bundles. Encourage students to count by tens to find
the total and add on any remaining popsticks. Students should then label
the collection using numeral cards. Interlocking blocks, such as Multilink,
Unifix or Centicubes, could also be used.
Trading game
Counting on
Prepare numeral cards in the range eleven to nineteen and place them
face down on the floor. Provide the students with two collections of
counters. One collection should consist of bundles of ten counters, all of
the same colour. The second collection should consist of single counters
of assorted colours. Students take turns to select a card. They then collect
a corresponding number of counters, using the bundles of ten and single
counters. Encourage students to count on from the bundle of ten. This
activity may be varied by extending the range of numbers or by using ten
strips (made of ten dots on strips of card) instead of counters.
Why?
Developing an understanding of tens and ones will assist students in using
strategies other than counting by ones to solve problems.
151
FIGURATIVE
Supply students with a collection of base ten material. The students take
turns to throw a die and take a corresponding number of base ten “shorts”
from a central pile. On succeeding throws of the die, students add
appropriate numbers of “shorts” to their collection. As the students collect
ten “shorts” they swap or trade them for one base ten “long”. Continue
the activity until one, or all students, can trade ten “longs” for a base ten
“flat”.
Developing efficient numeracy strategies
Where are they now?
FIGURATIVE
Students do not see ten as a countable
unit. The student’s focus is on the
individual items that form a unit of ten.
In addition or subtraction tasks involving
tens, students count forwards and
backwards by ones.
Where to next?
Students see ten as a unit composed of
ten ones and are able to use the unit to
count. However, they are dependent on
visual representations of ten, such as
strips of card showing ten dots or using
ten fingers.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
WMS1.1 Asks questions that could be explored using mathematics in relation to
Stage 1 content
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Base ten: level 1.
152
How?
Popsticks
FIGURATIVE
Make a base board by folding a piece of paper or cardboard in half to form
two columns. Label the columns as “units” and “tens”. Construct a set of
numeral cards for the range one to nine on white cards. These cards will
be used to represent numerals in the “tens” column on the chart. Construct
a second set of numeral cards in the range zero to nine on coloured card.
These will be used to represent numerals in the “ones” column . Provide
bundles of ten white popsticks and a pile of coloured popsticks. Shuffle
the two decks of cards separately. Place the cards face down between the
students. The students take turns to turn over a white card and a blue
card to form a two-digit numeral and place the cards onto the chart. The
students then read the numeral they have formed and collect a
corresponding number of sticks, using the bundles of white popsticks
and the coloured popsticks. Students then place the popsticks next to the
numeral cards and allow others to verify that the number of popsticks
used is correct.
Why?
Developing an understanding of tens and ones will assist students in using
strategies other than counting by ones to solve problems.
153
Developing efficient numeracy strategies
Where are they now?
Students do not see ten as a countable
unit. The students’ focus is on the
individual items that form a unit of ten.
Where to next?
FIGURATIVE
Students see ten as a unit composed of
ten ones and are able to use the unit of
ten when counting, aided by visual
representations of ten, such as ten
strips.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Backward number word sequence
Forward number word sequence
Numeral identification
Base ten strategies: intermediate concept of ten.
154
How?
Flip and see
Provide each student with a large collection of popsticks and a base board
divided into a “tens” and a “ones” column. Place numeral cards in the
range zero to nine face down on the floor. The students take turns to flip
over two numeral cards and place one card in the tens column and one
card in the ones column on their base board. Students then bundle
popsticks into tens and place the correct number of bundles and units
onto their base board to match the numeral cards. Discuss how many
tens and ones were made.
•
Students complete the above activity and then swap the numeral
cards from the tens column to the units column and vice-versa.
They then repeat the activity.
•
Construct two sets of numeral cards in the range zero to nine. Flip
over two numeral cards and ask the students to select identical
numeral cards from the second set of cards. Ask students to place
their cards in the tens and ones column so that they form the
largest and the smallest number possible.
•
Organise students into pairs and provide each pair of students with
a set of numeral cards in the range zero to nine. The students
shuffle the cards and place them face down on the floor. They then
take turns to select two numeral cards. Using the two cards selected,
each student forms the largest two-digit number possible. The two
students then compare their numbers and the player with the larger
number scores ten points. Continue playing until one player gains
a score of one hundred.
Why?
Developing an understanding of tens and ones will assist students in using
strategies other than counting by ones to solve problems.
155
FIGURATIVE
Variations
Developing efficient numeracy strategies
Where are they now?
Students:
• are able to automatically represent a
number on their fingers. They can
usually produce the number word after
a given number word in the range of
one to twenty.
Where to next?
• are able to find the total of two groups
of objects without touching the objects
but need to start from “one”.
• are able to count on from a given
number to find the total of two
groups
Students:
• automatically produce the number
word after a given number within the
range of one to thirty
FIGURATIVE
• are able to count forwards to thirty
and backwards from thirty or beyond.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.1 Asks questions that could be explored using mathematics in relation to Stage
1 content
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems.
LFN reference
Forward number word sequence
Backward number word sequence
Counting on
156
How?
Bucket count on
Drop a small collection of blocks one by one, into a bucket. Ask students
to count aloud as each block is added to the container. After dropping the
blocks, show the students the contents of the bucket. Then hold the bucket
above the eye level of the students. Ask the students to state how many
blocks would be in the bucket if one more block was added. Repeat the
question, changing the number of blocks to be added to two and three
blocks. Encourage the students to count on from the number of blocks
already in the bucket to find the total.
Variation
FIGURATIVE
Ask the students to pretend there are a nominated number of blocks in
the bucket. Drop additional blocks into the bucket. Students count on to
find the total sum of the blocks in the bucket.
Why?
These activities encourage students to count on from a given number
and assist in developing their knowledge of the forward sequence of
number words.
157
Developing efficient numeracy strategies
Where are they now?
Students:
• can produce the number word that
follows a given number word, in the
range of one to twenty
• can say the backward number word
sequence from “twenty” to “one”
• can say the number word just before a
given number word, but may drop
back to counting from “one” when
doing so.
Where to next?
Students:
• are able to count on from a given
number
• automatically recall number facts for
ten
FIGURATIVE
• automatically say the number word
before or after a given number.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.1 Asks questions that could be explored using mathematics in relation to Stage
1 content.
LFN reference
Figurative counting
Base ten strategies
Quinary strategies
158
How?
Blocks on a bowl
FIGURATIVE
Place a container, such as an empty ice-cream container, between a pair
of students. Turn the container upside down and place five Unifix blocks
on top. Instruct students to look away while their partner takes away
some, or all, of the blocks from the top of the container and hides them
under the container. The first student turns back to see how many blocks
are left on top of the container. Using this information, the student
determines how many blocks were placed under the container. The student
may then lift the container to confirm the answer.
Variation
As students become competent with five blocks, ten and then twenty blocks
could be used.
Why?
Providing students with opportunities to solve tasks involving hidden or
screened items may encourage them to use “counting on” as a problem
solving strategy.
159
Developing efficient numeracy strategies
Where are they now?
Students :
• are able to identify numerals in the
range of one to twenty
• are competent in saying the forward
number word sequence to twenty.
Where to next?
Students:
• are able to identify one- and twodigit numerals
• are able to say the forward number
word sequence to 100.
FIGURATIVE
Ensure squares on the 10 by 10 grid
are large enough to easily
accommodate the size of a counter.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
PAS1.1
Creates, represents and continues a variety of number patterns, supplies
missing elements in a pattern and builds number relationships
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.3 Describes mathematical situations and methods using everyday language and
some mathematical language, actions, materials, diagrams and symbols.
LFN reference
Numeral identification
160
How?
Hundred chart
Display a large hundred chart. Ask students to identify, and explain to the
group, any number patterns they can see on the hundred chart. After
practice with the large hundred chart, give the students small hundred
charts to work with. Cut the charts into strips so that numerals are grouped
into tens. Students then sequence these numeral strips to recreate the
hundred chart. Students then use these hundred charts to discover and
record their own number patterns. Provide the students with calculators
to confirm the number patterns.
•
Provide students with a 10 by 10 grid to represent a hundred chart
and fifteen counters, each displaying a different numeral in the
range of one to one hundred. Students place the counters onto
the grid in the correct numerical position. It may be necessary to
provide a “key” numeral amongst the fifteen counters, for example
the numeral 50.
•
Cut groups of numerals on a hundred chart into a square formation.
For example, cut the chart so that the numerals 1, 2, 11 and 12 are
together. Alternatively, cut the hundred chart in a random pattern
similar to a jigsaw design. Students then restore the hundred
charts.
•
Display a blank hundred chart on an overhead projector. Students
call out numbers up to one hundred. The teacher, or a student,
writes the nominated numerals onto the chart as they are named.
•
Provide students with individual hundred charts. Blank out some
of the numerals on the chart. Have the students write the missing
numerals in the spaces.
•
Allow opportunities for students to practise oral counting forwards
and backwards as well as skip counting by twos, fives and tens.
Why?
These activities assist students in developing their knowledge of the relative
size of numbers to 100.
161
FIGURATIVE
Variations
Developing efficient numeracy strategies
Where are they now?
Students are able to complete addition
tasks, but count from “one” to find the
total, rather than counting on from the
larger group.
Where to next?
FIGURATIVE
Students use counting on as a strategy to
solve addition problems.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.3 Describes mathematical situations and methods using everyday and some
mathematical language, actions, materials, diagrams and symbols
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained
LFN reference
Counting on
162
How?
Add two dice
FIGURATIVE
Construct a set of numeral cards in the range of two to twelve. Place
them face up on a table, or on the floor. Taking turns, the students are to
roll two dice and find the total. Encourage the students to count on from
the larger number rolled. After adding the two dice the student takes a
numeral card corresponding to the total. The game continues until all
the cards have been taken. If a player rolls a number that has already been
taken, the player’s turn is forfeited.
Why?
Students need to be encouraged to use increasingly efficient strategies to
solve addition tasks.
163
Developing efficient numeracy strategies
Where are they now?
Students are able to complete addition
tasks, but count from “one” to find the
total, rather than counting on from the
larger group.
Where to next?
FIGURATIVE
Students use counting on as a strategy to
solve addition problems.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.3 Describes mathematical situations and methods using everyday language and
some mathematical language, actions, materials, diagrams and symbols.
LFN reference
Counting on
164
(Add two dice)
•
Use a variety of dice to extend the range of numbers. For example,
construct a dodecahedron (12 faces) from plastic polygon shapes
and attach numerals to each face of the construction. Modify the
set of numeral cards to the appropriate range of numbers.
•
Provide the students with three dice and with numeral cards for
three to eighteen. This activity is then played like Add two dice.
Students roll the three dice and find the total. This provides
opportunities for introducing strategies other than counting by
ones to solve addition tasks.
•
Allow students to construct their own die and attach numerals of
their choice. If large numbers are written on the first die, then
modify the second die to display only the numerals 1, 2 and 3. A
calculator may be used to confirm the additions.
•
Another variation of the activity is achieved by instructing the
students to write five numbers, within a nominated range, on a
strip of paper. Students take turns to roll two dice and find the
total. They then tell the group the total. As the totals are called,
students cross off any corresponding numerals on their paper strip.
The game continues until one student has crossed off all five
numerals on his or her paper.
Why?
Students need to be encouraged to use increasingly efficient strategies to
solve addition tasks.
165
FIGURATIVE
Variations
Developing efficient numeracy strategies
Where are they now?
Students are able to solve addition
problems up to ten or beyond, but they
count from one to do so.
Where to next?
FIGURATIVE
Students automatically recall addition
number facts to ten.
Outcomes
This activity provides opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.3 Describes mathematical situations and methods using everyday language
and some mathematical language, actions, materials, diagrams and
symbols.
LFN reference
Figurative counting
Finger patterns
166
How?
Rabbits’ ears
Instruct the students to make two fists and rest them on their heads, so
that their hands are out of their direct line of sight. Ask the students to
raise a given number of fingers on each hand and to add them together.
Students may bring their hands down to confirm the answer.
Doubles
Instruct the students to use two hands to demonstrate double numbers
from 1 to 5. For example, “Show me double four. How many altogether?”
In this example the students would raise four fingers on each hand and
call out the answer. Students may bring their hands down to count and
confirm the total.
This activity is played in a similar way to Doubles. Instruct the students
to raise their fingers for a nominated double combination and then add
one more finger to find the total. Alternatively, play Doubles minus one.
For this activity students raise their fingers to represent a nominated
double and then subtract one finger to find the total.
Why?
Students use finger patterns to support their early understanding of
numbers. Understanding that numbers are made up of other numbers
allows students to move to strategies which don’t involve counting by
one. Using doubles and near doubles is a common early use of number
facts.
167
FIGURATIVE
Doubles plus one
Developing efficient numeracy strategies
Where are they now?
Students:
• instantly recognise die patterns to “six”
• are able to add two screened groups
after being told the amount in each
group, but count the total starting
from “one”.
Where to next?
Students:
• use the strategy of counting on from
the larger group when adding two
groups to complete addition tasks
• recall number facts to “ten”.
FIGURATIVE
Encourage students to predict
which number will need to be to
found to make a total of ten, before
turning over the second card.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Counting on
Subitising
168
How?
Dot pattern cards
FIGURATIVE
Construct dot patterns for numbers one to nine, with two copies of card
5. Place the cards face down on a table between pairs of students. The
students take turns to turn over two cards and add the two cards together.
If the total is “ten” the student keeps the two cards. If the cards do not
equal “ten” they are returned to the table. Encourage students to count
on from the larger number.
Variations
•
Use numeral cards instead of dot pattern cards.
•
Use five numeral cards and five dot pattern cards.
•
Use ten frame cards.
•
Extend the total to 15 or 20.
Why?
Students need to develop a range of strategies to solve number problems.
Strategies such as “counting on”, “counting down to” and “counting up
to” may be an efficient method of solving a problem. However this is
dependent upon the task and the student knowing which is the most
appropriate strategy.
169
Developing efficient numeracy strategies
Where are they now?
When adding two groups, the students
count from “one” to find the total.
Where to next?
FIGURATIVE
Students use the strategy of counting on
from the larger group when adding two
groups.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Counting on
170
How?
Posting counters
Provide the students with a container similar to a money box. Instruct
the students to “post” a nominated number of counters through the slot
in the container. Encourage the students to count each counter as it is
dropped through the slot. Students then pause and state the total number
of counters that are now hidden in the container. Instruct the students to
post an additional number of counters into the container, counting on
from the original number. Alternatively, ask students to pretend there
are a specified number of counters in the container. Students count on as
additional counters are posted through the slot.
Two-dice toss
FIGURATIVE
Provide each pair of students with two dice and a pile of counters. Have
the students take turns to throw the dice and add the total, counting on
from the larger number. Students then take the corresponding number
of counters from a central pile. The game continues until all the counters
have been removed from the central pile.
Why?
Students need to develop strategies other than counting all items in
groups, starting from one, when solving addition tasks. Counting on is a
more efficient strategy.
171
Developing efficient numeracy strategies
Where are they now?
Students add two groups by counting
from “one” rather than counting on
from the larger group.
Where to next?
Students use the strategy of counting on
from the larger of two groups to solve
addition tasks.
FIGURATIVE
Teachers may take this opportunity
to model addition using counting
on procedures.
Outcomes
These activities provide opportunities to demonstrate progress towards the following
outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
PAS1.1
Creates, represents and continues a variety of number patterns, supplies
missing elements in a pattern and builds number relationships
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.3 Describes mathematical situations and methods using everyday language
and some mathematical language, actions, materials, diagrams and
symbols.
LFN reference
Counting on
Combining procedures
172
How?
Teddy tummies: addition
Variations
•
The students record the number combinations for “ten” as they
are discovered. Allow opportunities for the students to demonstrate
their “discoveries” to the rest of the class.
•
Substitute the “teddies” blackline master with BLM on page 213
displaying two trucks. Provide pairs of students with ten plastic
teddies to complete the activity in a similar way to Teddy tummies.
Why?
Students need opportunities to use counting on as an addition strategy
and to develop knowledge of the numbers that combine to make ten.
173
FIGURATIVE
Prepare base cards using BLM on page 212. Provide each pair of students
with ten counters and the appropriate base board. The students take turns
to distribute the counters between the two teddies by placing the counters
on the teddies’ tummies. Pairs of students then discuss the number
combinations formed with the ten counters. The students continue the
activity, investigating the possible number combinations for the ten
counters.
Developing efficient numeracy strategies
Where are they now?
FIGURATIVE
Students demonstrate their
understanding of the process of addition
by joining two groups. They are able to
find the total of the two groups by
counting all of the items, starting from
“one”.
Where to next?
Students automatically recall number
facts to 10.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
PAS1.1
Creates, represents and continues a variety of number patterns, supplies
missing elements in a pattern and builds number relationships
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.3 Describes mathematical situations and methods using everyday language
and some mathematical language, actions, materials, diagrams and symbols.
LFN reference
Counting on
Partitioning and combining strategies
174
How?
Friends of ten
FIGURATIVE
Construct two sets of numeral cards in the range of one to ten. For this
activity it is necessary to attach string or shoelaces to the numeral cards
so they can be worn around the students’ necks. It is also more manageable
if each set of cards is a different colour. Distribute one set of numeral
cards to ten students. These students leave the room or turn away from
the remaining students. Distribute the other set of numeral cards to the
remaining students. Ask the students in the first group to return to the
class (or turn around) and find a partner who is wearing a card which,
when added to their own card, will equal ten.
Variations
•
Increase the range of numbers on the numeral cards.
•
Change the cards so that one set displays numerals and the other
set displays dot patterns.
Why?
Knowing the basic number combinations that form ten allows students to
use a range of strategies for addition, for example, knowing that 7+3 is the
same as 8+2, and introduces the idea of compensation (one up, one down).
This activity develops the concept that addition may be an appropriate strategy
for solving a subtraction problem.
175
Developing efficient numeracy strategies
Where are they now?
FIGURATIVE
Students demonstrate their
understanding of the process of addition
by joining two groups. They are able to
find the total of the two groups by
counting all of the items, starting from
“one”.
Where to next?
Students automatically recall number
facts to 10.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
PAS1.1
Creates, represents and continues a variety of number patterns, supplies
missing elements in a pattern and builds number relationships
WMS1.3 Describes mathematical situations and methods using everyday and some
mathematical language, actions, materials, diagrams and symbols
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained
LFN reference
Counting on
Partitioning and combining strategies
176
Fishing addition
4+1 2+3
3 +33+4 2 + 4
3+3 2+4
3+3 2+4
3 + 3 72 + 4
+
0
3+3 2+4
5+0
25++51
5+1
5+1
56++11
5+1
3+2
6+0
6+0
6+0
6+0
6+0
Why?
Knowing basic number combinations that form ten allows students to use a
range of strategies for addition, for example, knowing that 7+3 is the same
as 8+2, and introduces the idea of compensation (one up, one down).
This activity develops the concept that addition may be an appropriate strategy
for solving a subtraction problem.
177
FIGURATIVE
Construct six sets of four cards, with each card in any one set displaying
a basic addition fact for the same number in the range one to ten. Each
set of cards should have a unique sum. Shuffle the cards and deal five
cards to each player. Place the remainder of the cards face down on the
floor. Instruct the students to look at their dealt cards and take turns to
discard any pairs of cards that add up to the same total. After discarding
any pairs of cards, the first player asks his or her partner for a card which
equals a specific number. For example: “Pass me an eight”. If the partner
holds a card totalling the nominated number, it must be handed over. If
the partner does not have the nominated card, the person who asked
takes a card from the central pile. In both cases, if a pair of cards is formed,
the student discards them. The winner is the player who discards all of
his or her cards first.
Developing efficient numeracy strategies
Where are they now?
Students:
• can identify numerals from one to
twenty
• are able to say the forward number
word sequence to twenty
• confidently repeat the backward
number word sequence from twenty.
Where to next?
Students:
• can identify numerals to 100 or beyond
FIGURATIVE
• are able to say the forward number
word sequence to 100 or beyond,
confidently crossing into the next
group of ten numbers.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Numeral identification: level 3
178
How?
Using technology
Provide the students with at least one calculator for each pair of students.
Instruct the students to:
•
enter a given number on the calculator
•
enter on the calculator the number which is one before or one
after a given number
•
use the constant function on a calculator to add on from a given
number.
FIGURATIVE
Provide the students with a hundred chart on which they can record the
number patterns.
Why?
This activity provides an opportunity for students to develop their
knowledge of numbers up to and perhaps beyond 100.
179
Developing efficient numeracy strategies
Where are they now?
Students can solve addition tasks but
count from one to find the total of two
groups. Students are able to solve
subtraction tasks when concrete
material is available.
Where to next?
Students are able to complete addition
and subtraction tasks using the
strategies of “counting on” and
“counting down from”.
FIGURATIVE
Allow students to discover that
some numbers cannot be
subtracted if they do not have
enough counters to begin with.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.3 Describes mathematical situations and methods using everyday and some
mathematical language, actions, materials, diagrams and symbols
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Combining and partitioning
180
How?
Ring that bell
Provide the students with a supply of counters. Ring a bell a number of
times, for example, four times. Instruct the students to place the
corresponding number of counters on their desk. Hold up a symbol card
for addition or subtraction and then ring the bell again, for example twice.
Have the students respond by observing the symbol card and adding or
subtracting the correct number of counters. Students then state the total
number of counters. Encourage the students to discuss their actions and
how they arrived at their answers.
Variation
FIGURATIVE
This may be used as a small-group or partner activity, with students rolling
a die instead of ringing a bell.
Why?
Students need opportunities to use more efficient number strategies when
completing addition and subtraction tasks.
181
Developing efficient numeracy strategies
Where are they now?
Students are able to find the total of
two groups by counting from “one”.
Where to next?
Students:
• use a counting on strategy to solve
addition tasks
• count in a forward and backward
sequence to 100 and beyond
• have automatic recall of addition and
subtraction facts to ten.
FIGURATIVE
Introduce the activity to the whole
class before students complete the
activity in small groups or with
partners.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Figurative counting
Combining
Spatial patterns
Forward number word sequence
Backward number word sequence
182
How?
How many eggs?
Construct a grid using BLM on page 209 as well as one set of rule cards
and a set of numeral cards for each group of students. Rule cards need to
display either an addition or subtraction sign, followed by the numeral 1,
2 or 3. For example, a rule card might display: “ + 3”.
FIGURATIVE
Instruct the students to place the correct number of counters on each
column according to the numeral written at the top of the column.
Students then take turns to turn over a rule card from the pile. Students
follow the rule, to add or subtract counters from each column and
determine the new total. As the students determine each total, they place
a corresponding numeral card at the bottom of each column.
Variation
Modify the numerals at the top of each column.
Why?
This activity provides an opportunity for students to develop awareness
of number patterns and to develop strategies for solving problems other
than counting by ones.
183
Developing efficient numeracy strategies
Where are they now?
Students are able to solve addition tasks,
but count from one to find the total of
the two groups.
Where to next?
Students complete addition or missing
addend tasks using the strategies of
counting on or counting back.
FIGURATIVE
This activity can be demonstrated
by the teacher using an overhead
projector and transparent counters.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Figurative counting
184
How?
Ten frames
Provide each student with a blank ten frame and ten counters. Ask students
to place a specified number, less than ten, on their ten frame. Direct
students to add another specified number of counters to the frame and to
find the total.
Variations
Complete subtraction tasks using the ten frames. For example,
arrange nine counters on the ten frame. Ask students to determine
how many counters would need to be taken away to leave six.
Demonstrate the strategies of counting down to and counting down
from a given number.
•
Use two ten frames to work with numbers to twenty.
FIGURATIVE
•
Why?
Ten frames provide students with a visual structure for a number. This
encourages students to establish and work with visual images of numbers.
Ten frames emphasise doubles and five as part of a number. For example,
using a ten frame, 9 can be seen as 5+4, one less than double 5 or one
more than double 4.
185
Developing efficient numeracy strategies
Where are they now?
Students:
• understand and demonstrate the
meaning of subtraction by taking an
object, or a group of objects, from a
group of objects.
• are able to count in a backward number
word sequence from twenty.
Where to next?
Students are able to use addition or to
count down from a number to find the
answer to a subtraction problem.
FIGURATIVE
A full explanation of the strategies
of Counting down to and Counting
down from appear on the opposite
page.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.2
Uses a range of mental strategies and informal recording methods for
addition and subtraction involving one- and two-digit numbers
WMS1.3 Describes mathematical situations and methods using everyday and some
mathematical language, actions, materials, diagrams and symbols
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Figurative counting
186
How?
Subtraction teddies
Counting down to
The student counts backwards from the larger number when solving
subtraction problems where the problem involves a missing addend. For
example, when solving 9 - ( ) = 6, the students would count backwards
from nine knowing they are counting to the number six and say “eight,
seven, six.” Students typically hold up fingers as they count and recognise
three as the answer.
Counting down from
The student counts backwards from the larger number when solving
subtraction problems. For example, when solving 9 - 3, the student counts
backwards from nine saying “eight, seven, six...six!”
Why?
This activity is designed to develop students’ knowledge of subtraction
facts. Automatic recall of addition and subtraction facts allows students
to attend to other features when solving problems.
187
FIGURATIVE
Provide each student with twenty plastic teddies, a “double decker bus”
baseboard (see BLM on page 214) and a strip of paper. Have the students
place the twenty teddies on the bus baseboard. Instruct the students to
take turns to roll a die and subtract the corresponding number of teddies
from the collection of teddies on the bus. The student then records the
number of remaining teddies on the strip of paper. The activity continues
until one student reaches zero.
Developing efficient numeracy strategies
Where are they now?
Students count from “one” to solve tasks
involving equal multiples.
Where to next?
Students are able to solve multiplication
tasks by counting in multiples.
FIGURATIVE
Encourage students to discuss how
they reached their answer. Allow
them to verbalise how they counted.
If the students are unable to skip
count by threes, encourage
rhythmic or stressed counting.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication and division
188
How?
Arrays
Organise students into pairs. Provide the students with a collection of
counters. Instruct one of the pair to make a simple array that is no larger
than 5 by 5, with counters. The student then briefly shows the array to
his or her partner before screening the counters with a sheet of cardboard.
Arrays: changing groups
Arrange nine students into three rows with three students in each row.
Pose the following question: “If we add another row of children, how many
would there be altogether?”
Continue adding rows of students and encourage students to guess how
many children there are altogether, prior to counting the students.
Variation
Change the number of students in each line.
Why?
Early multiplication and division strategies focus on the structure and
use of groups of items. Students need to develop strategies where they
see a group of items as one unit and no longer need to count each item
within the group.
189
FIGURATIVE
The other student then attempts to construct the same array pattern with
counters. The students should then compare the two arrays. Ask the
students to find the total number of counters in the array.
Developing efficient numeracy strategies
Where are they now?
Students need visible items to calculate
the total number of items in a simple
square array pattern.
Where to next?
Students are able to calculate the total
number of items in a simple square array
pattern without relying on visual
representations.
FIGURATIVE
Observe and question the students
to determine how they are
calculating the total in the array.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication
190
How?
Guess my square
FIGURATIVE
Organise students into pairs so that they are sitting opposite each other.
Provide each student with an equal number of multi-link cubes or pattern
tiles and a sheet of cardboard which will be used as a screen. Ask students
to take turns to construct a simple array pattern using the material
provided. Students should screen the array from their partner until the
pattern is completed. The student then briefly shows the array to the
partner before screening the array again. The other student constructs
the same array pattern from memory. Instruct students to uncover and
compare both array patterns. The students then calculate the total number
of items in each array.
Variation
Arrange students in pairs. Ask the students to take turns to make an array
which is hidden from their partner’s view. The student then describes the
array to the partner by stating the number of counters in each row and
the number of rows. The partner then attempts to make the array. He or
she then determines the total number of counters in the array.
Why
The formation of arrays is important in developing strategies of
coordinating groups. This coordination of groups can lead to students
developing abstract concepts to solve multiplication and division problems.
191
Developing efficient numeracy strategies
Where are they now?
The students determine the total
number of items within a specified
number of groups by counting the total
by “ones”.
Where to next?
FIGURATIVE
Students use the strategies of rhythmic
counting and skip counting to find the
total number of items within a specified
number of groups.
Literature link
Many traditional folk and fairy tales
have a theme based on three, such
as The Three Little Pigs and
Goldilocks and the Three Bears.
Grouping based on these tales can
be used in activities.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
PAS1.1
Creates, represents and continues a variety of number patterns, supplies
missing elements in a pattern and builds number relationships
WMS1.3 Describes mathematical situations and methods using everyday and some
mathematical language, actions, materials, diagrams and symbols
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication and division
192
How?
Triangle teddies
FIGURATIVE
Provide the students with a collection of popsticks and a collection of
plastic teddies. Instruct the students to make a triangle using three sticks.
Ask the students to then place a teddy on each of the corners of the triangle.
The students then count and record the number of teddies on the triangle.
Have the students repeat the process for a second triangle. The students
record the total number of teddies placed on the two triangles. Have the
students continue to form additional triangles and record the total number
of teddies on the triangles.
Why?
Students need to be able to recall the counting sequences for nominated
multiples. Multiplication and division strategies will be limited by students’
knowledge of these sequences.
193
Developing efficient numeracy strategies
Where are they now?
The students determine the total
number of items within a specified
number of groups by counting the total
by “ones”.
Where to next?
FIGURATIVE
Students use the strategies of rhythmic
counting and skip counting to find the
total number of items within a specified
number of groups.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.1
Counts, orders, reads and represents two- and three-digit numbers
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication and division
194
How?
Units for two
Collect or draw items commonly found in pairs, such as eyes, shoes, socks,
or legs. Model the method of counting the items, using rhythmic counting,
to the students. Model methods for keeping track of the number of groups
as well as the total number of items. Allow opportunities for the students
to practise the modelled methods. Demonstrate how the total number of
items in a specified number of “groups of two” can be found by counting
the first number in each pair silently and voicing the second number.
Allow opportunities for the students to practise this counting method.
Handprints
Make a handprint to represent a group of five. Repeat printing the
handprints across a strip of paper. Ask the students to count the number
of hands and to find the total number of fingers using rhythmic or skip
counting methods.
Variation
FIGURATIVE
Construct a square using four popsticks to represent groups of four.
Why?
Students need to be able to recall the counting sequences for nominated
multiples. Multiplication and division strategies will be limited by students’
knowledge of these sequences.
195
Developing efficient numeracy strategies
Where are they now?
Students are able to form equal groups of
items but find the total number of items
in the groups by counting by ones.
Where to next?
Students are able to use rhythmic and
skip counting strategies to find the total
of equal groups.
FIGURATIVE
Write the instruction cards in two
ways, such as four rows of 5 and five
rows of 4.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication and division: level 2
196
How?
Turning arrays
FIGURATIVE
Provide each student with a small sheet of cardboard and a supply of
counters. Instruct students to form arrays by placing the counters onto
the cardboard following instructions, such as “make three rows of five
counters”. Students then turn the card 90° to show a new array of five
rows of three. Discuss with the students the number of rows, the number
of counters in each row and the total number of counters for each array
pattern.
Variation
Allow the students to form arrays using potato prints, shape prints, thumb
prints or adhesive stickers. Provide students with instruction cards for
making the arrays.
Why
The formation of arrays is important in developing concepts of groups
and coordinating groups. This coordination of groups can lead to abstract
concepts of multiplication and division.
197
Developing efficient numeracy strategies
Where are they now?
Students are able to form items into
equal groups but count by ones to find
the total.
Where to next?
Students are able to use the strategies of
rhythmic and skip counting to find the
total of equal groups.
FIGURATIVE
BLMs could be covered with contact
and felt pens used to circle the
groups. They can then be erased for
later use by others in the class.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication and division: stages 1, 2
198
How?
Calculating groups
Provide students with a stencil displaying a large number of items. Instruct
students to draw rings around groups of items on the stencil. For example:
“Put a ring around groups of five items.” Ask the students to determine
how many groups were made and to use rhythmic counting or skip
counting to find the total. Ask the students to determine the number of
single items remaining on the sheet.
Construct multiple copies of the grid on BLM on pages 210 and 211 and
accompanying picture cards displaying groups of items. Each pair of
students will need a copy of the grid, a deck of picture cards and counters
of two different colours. Shuffle the picture cards and place them face
down before the students. The first player takes a card from the top of the
pile and places a counter on the corresponding square on the grid. For
example, if a card displaying one item is drawn, the student places his or
her counter on the “one group of 1” square at the top left-hand corner of
the grid. Players continue to take turns to turn over cards and mark them
on the grid. The winner is the first player to make a line of three counters
horizontally, vertically or diagonally.
To extend this activity introduce numeral cards which indicate the total
number of items on each picture card. After the students place a picture
card onto the grid, instruct them to determine the total of the groups and
place a corresponding numeral card on top of the picture card.
Why?
Early multiplication and division strategies focus on the structure and
use of groups of items. Students need to develop the concept of seeing a
group of items as one “unit” and no longer relying on counting each item
within the group.
199
FIGURATIVE
What’s in a square?
Developing efficient numeracy strategies
Where are they now?
Students are able to form items into
equal groups. They calculate the total
number of items in the groups by
counting by ones.
Where to next?
FIGURATIVE
Students are able to use the strategies of
rhythmic and skip counting to find the
total of equal groups.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication and division: stage 2
200
How?
Groupies
Provide the students with a collection of items such as counters, marbles
or plastic teddies. Direct the students to form a specified number of equal
groups using the items, for example, “Make four groups of three counters”.
Students use rhythmic or skip counting to find the total number of items
in the specified groups.
Variation
FIGURATIVE
Use dot patterns on dice to represent groups, for example, five groups of
two.
Why?
Early multiplication and division strategies focus on the structure and
use of groups of items. Students need to develop the concept of seeing a
group of items as one “unit” and no longer relying on counting each item
within the group.
201
Developing efficient numeracy strategies
Where are they now?
Students are able to form items into
equal groups. They calculate the total
number of items in the groups by
counting by ones.
Where to next?
FIGURATIVE
Students are able to use the strategies of
rhythmic and skip counting to find the
total of equal groups.
Outcomes
These activities provide opportunities for students to demonstrate progress towards the
following outcomes: A student
NS1.3
Uses a range of mental strategies and concrete materials for multiplication
and division
PAS1.1
Creates, represents and continues a variety of number patterns, supplies
missing elements in a pattern and builds number relationships
WMS1.2 Uses objects, diagrams, imagery and technology to explore mathematical
problems
WMS1.4 Supports conclusions by explaining or demonstrating how answers were
obtained.
LFN reference
Early multiplication and division: stage 2
202
Dice and grid game
Construct a 6x6 grid for each player. Write the numerals 1 to 36, in
sequence, onto the grid, placing one numeral in each square. Each player
will also need six sets of dot pattern cards representing the numbers one
to six, a numeral die and a dot pattern die.
In this activity, the “numeral die” is used to indicate the number of equal
groups, and the “dot pattern die” indicates the number of items in each
group.
Instruct the students to roll the two dice and state the size of each group
and the number of equal groups. When the student has stated the size
and number of equal groups, ask the student to collect the correct number
of dot pattern cards to represent the groups. For example, if the student
wishes to make five groups of three, he or she would then select five cards
showing three dots on each card.
Variation
Allow students to use a calculator to determine the total number of dots
in the groups. Observe whether the student uses repeated addition or
multiplication with the calculator.
Why?
Early multiplication and division strategies focus on the structure and
use of groups of items. Students need to develop the concept of seeing a
group of items as one “unit” and no longer relying on counting each item
within the group.
203
FIGURATIVE
Students then find the total number of dots by rhythmic or skip counting,
and place a counter on the corresponding numeral on the grid. In the
above example, the student would place the counter on the numeral 15.
The winner is the first to have three counters in a row, horizontally,
vertically or diagonally.
Developing efficient numeracy strategies
FIGURATIVE
Assessment tasks
Task
Student response
Assessment
T :”If I have 6 pencils
and I get another 3,
how many do I have
altogether?”
S: Correctly adds the
two groups without the
use of concrete
material.
Did the students count
from one, or did they
count on from 6?
T: Displays a set
of eight counters.
“Here are eight
counters.”
Screens the counters.
The teacher then
displays a set of 5
counters.
“There are five
counters here”.
Screens the second
group.
“How many counters
are there altogether?”
S: Demonstrates
addition by counting on
from the larger
number.
Did the student:
• count from one?
• count on from eight?
• know the number fact
automatically?
If the student was
unsuccessful in adding
the two groups, the
teacher should
unscreen the second
group and pose the
question again. This
may encourage the
student to count on.
T: “If I have 8
lollies and I eat 3, how
many are left?”
S: Completes the
subtraction task without
the use of concrete
materials.
Which strategy did the
student use to
determine the answer?
T: Displays and then
screens a collection
of 15 counters.
“I’m taking out 6. How
many are left?”
Displays the six removed
counters.
S: Completes the
subtraction task without
having to see or feel
the counters.
Did the student
• count on from 6?
• attempt to count down
from 15?
• attempt to count down
to 6?
T: Displays numeral cards
in the range from
1 to 100, and asks the
student to name the
numeral being
displayed.
S: Correctly names
each numeral as the
cards are displayed.
Is the student able to
name numerals in the
range:
1-10?
1-20?
1-100?
T: “Count forwards
starting at 55” (and stop
at 64).
S: Correctly
says the forward
number word sequence
from 55 to 64.
Does the student:
• know the forward
number word
sequence?
• count fluently from
one decade to the
next? (e.g. 59, 60)
204
Task
Student response
Assessment
T: “Count backwards
starting at 93” (and
stop at 87).
S: Correctly
says the backward
number word sequence
from 93 to 87.
Does the student:
• know the backward
number word
sequence?
• count fluently from
one decade to the
next? (e.g. 90, 89)
T: “Name the
numeral which comes
after ” (numbers in the
range 1 to 100,
e.g. the number after 53)
S: Correctly says the
numeral which comes
after the numerals
given by the teacher.
Did the student
drop back and count up
to find the next
number?
T: “Name the numeral
which comes before” (a
given numeral in the
range from 1 to 100,
e.g. the number before
53 is 52)
S: Correctly says the
numeral which comes
before the numerals
given by the teacher.
Does the student know
the backward number
word sequence from
100 to 1?
Did the student drop
back to a lower decade
to determine the
answer?
T: Briefly
displays 7 counters
before screening them.
“Here are seven
counters. I’m
taking some of them out
and there are 2 left.
How many did I take
out?”
S: Determines the
missing addend without
seeing or feeling the
counters.
Does the student model
and count the concealed
items?
Did the student count
down from 7 or count
down to 2?
Did the student count
from one, three times?
T: Displays a set
of numeral fact cards
(addition facts to ten.)
“What does this say?”
“Can you work it out?”
“How did you do it?”
S: Automatically recalls
number facts or uses an
efficient strategy to
solve the task.
Did the student
automatically provide
the answer to the
number fact?
If not, what strategy did
the student use?
Did the student:
• count from one?
• count on from the
larger number?
FIGURATIVE
Assessment tasks
205
Developing efficient numeracy strategies
Three-minute lesson
breakers
1
FIGURATIVE
2
•
Oral counting using the hundred chart.
•
Think of a “secret number” and provide the students with clues for
guessing the “secret number”. Include simple and complex clues.
For example:
3
4
5
6
206
“ The secret number is two more than three.”
“ The secret number is more than twelve but less than nineteen.”
“ The secret number is an odd number between 10 and 20.”
•
Provide opportunities for the class to practise oral counting from a
number other than one. For example, instruct the students to count
from 36 to 50.
•
Using one week on a calendar as a visual aid, pose such problems as:
“Count the number of breakfasts or total meals a person would eat in
one week.”
“Count the number of days already spent at school this
week and the number of days left until the weekend.”